Clifford Algebras, Clifford Groups, and a Generalization of the ... · Chapter 1 Clifford Algebras, Clifford Groups, and the Groups Pin(n) and Spin(n) 1.1 Introduction: Rotations

arX

iv:0

805.

0311

v3 [

mat

h.G

M]

28

Sep

2014

Clifford Algebras, Clifford Groups,and a Generalization of the Quaternions:

The Pin and Spin Groups

Jean Gallier

Department of Computer and Information ScienceUniversity of Pennsylvania

Philadelphia, PA 19104, USAe-mail: [email protected]

September 30, 2014

http://arxiv.org/abs/0805.0311v3

2

3

Clifford Algebras, Clifford Groups,

and a Generalization of the Quaternions:The Pin and Spin Groups

Jean Gallier

Abstract: One of the main goals of these notes is to explain how rotations in Rn are inducedby the action of a certain group Spin(n) on Rn, in a way that generalizes the action of theunit complex numbers U(1) on R2, and the action of the unit quaternions SU(2) on R3

(i.e., the action is defined in terms of multiplication in a larger algebra containing both thegroup Spin(n) and R

n). The group Spin(n), called a spinor group, is defined as a certainsubgroup of units of an algebra Cln, the Clifford algebra associated with Rn.

Since the spinor groups are certain well chosen subgroups of units of Clifford algebras,it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups.These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including astudy of the structure of the Clifford algebra Clp,q associated with a nondegenerate symmetricbilinear form of signature (p, q) and culminating in the beautiful “8-periodicity theorem” ofElie Cartan and Raoul Bott (with proofs).

4

Contents

1 Clifford Algebras, Clifford Groups, Pin and Spin 7

1.1 Introduction: Rotations As Group Actions . . . . . . . . . . . . . . . . . . . 71.2 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Clifford Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 The Groups Pin(n) and Spin(n) . . . . . . . . . . . . . . . . . . . . . . . . 251.5 The Groups Pin(p, q) and Spin(p, q) . . . . . . . . . . . . . . . . . . . . . . 311.6 Periodicity of the Clifford Algebras Clp,q . . . . . . . . . . . . . . . . . . . . 331.7 The Complex Clifford Algebras Cl(n,C) . . . . . . . . . . . . . . . . . . . . 371.8 The Groups Pin(p, q) and Spin(p, q) as double covers . . . . . . . . . . . . . 381.9 More on the Topology of O(p, q) and SO(p, q) . . . . . . . . . . . . . . . . . 42

5

6 CONTENTS

Chapter 1

Clifford Algebras, Clifford Groups,

and the Groups Pin(n) and Spin(n)

1.1 Introduction: Rotations As Group Actions

The main goal of this chapter is to explain how rotations in Rn are induced by the actionof a certain group Spin(n) on Rn, in a way that generalizes the action of the unit complexnumbers U(1) on R2, and the action of the unit quaternions SU(2) on R3 (i.e., the actionis defined in terms of multiplication in a larger algebra containing both the group Spin(n)and R

n). The group Spin(n), called a spinor group, is defined as a certain subgroup of unitsof an algebra Cln, the Clifford algebra associated with Rn. Furthermore, for n ≥ 3, we arelucky, because the group Spin(n) is topologically simpler than the group SO(n). Indeed, forn ≥ 3, the group Spin(n) is simply connected (a fact that it not so easy to prove withoutsome machinery), whereas SO(n) is not simply connected. Intuitively speaking, SO(n) ismore twisted than Spin(n). In fact, we will see that Spin(n) is a double cover of SO(n).

Since the spinor groups are certain well chosen subroups of units of Clifford algebras, it isnecessary to investigate Clifford algebras to get a firm understanding of spinor groups. Thischapter provides a tutorial on Clifford algebra and the groups Spin and Pin, including astudy of the structure of the Clifford algebra Clp,q associated with a nondegenerate symmetricbilinear form of signature (p, q) and culminating in the beautiful “8-periodicity theorem” ofElie Cartan and Raoul Bott (with proofs). We also explain when Spin(p, q) is a double-cover of SO(p, q). The reader should be warned that a certain amount of algebraic (andtopological) background is expected. This being said, perseverant readers will be rewardedby being exposed to some beautiful and nontrivial concepts and results, including Elie Cartanand Raoul Bott “8-periodicity theorem.”

Going back to rotations as transformations induced by group actions, recall that if V isa vector space, a linear action (on the left) of a group G on V is a map α : G × V → Vsatisfying the following conditions, where, for simplicity of notation, we denote α(g, v) byg · v:

7

8 CHAPTER 1. CLIFFORD ALGEBRAS, CLIFFORD GROUPS, PIN AND SPIN

(1) g · (h · v) = (gh) · v, for all g, h ∈ G and v ∈ V ;

(2) 1 · v = v, for all v ∈ V , where 1 is the identity of the group G;

(3) The map v 7→ g · v is a linear isomorphism of V for every g ∈ G.

For example, the (multiplicative) group U(1) of unit complex numbers acts on R2 (byidentifying R2 and C) via complex multiplication: For every z = a + ib (with a2 + b2 = 1),for every (x, y) ∈ R2 (viewing (x, y) as the complex number x+ iy),

z · (x, y) = (ax− by, ay + bx).

Now, every unit complex number is of the form cos θ + i sin θ, and thus the above action ofz = cos θ+ i sin θ on R2 corresponds to the rotation of angle θ around the origin. In the casen = 2, the groups U(1) and SO(2) are isomorphic, but this is an exception.

To represent rotations in R3 and R4, we need the quaternions. For our purposes, it isconvenient to define the quaternions as certain 2 × 2 complex matrices. Let 1, i, j,k be thematrices

1 =

(1 00 1

), i =

(i 00 −i

), j =

(0 1−1 0

), k =

(0 ii 0

),

and let H be the set of all matrices of the form

X = a1+ bi + cj+ dk, a, b, c, d ∈ R.

Thus, every matrix in H is of the form

X =

(a+ ib c + id

−(c− id) a− ib

), a, b, c, d ∈ R.

The quaternions 1, i, j,k satisfy the famous identities discovered by Hamilton:

i2 = j2 = k2 = ijk = −1,

ij = −ji = k,

jk = −kj = i,

ki = −ik = j.

As a consequence, it can be verified that H is a skew field (a noncommutative field) calledthe quaternions . It is also a real vector space of dimension 4 with basis (1, i, j,k); thus as avector space, H is isomorphic to R4. The unit quaternions are the quaternions such that

det(X) = a2 + b2 + c2 + d2 = 1.

Given any quaternion X = a1 + bi+ cj + dk, the conjugate X of X is given by

X = a1− bi− cj− dk.

1.2. CLIFFORD ALGEBRAS 9

It is easy to check that the matrices associated with the unit quaternions are exactly thematrices in SU(2). Thus, we call SU(2) the group of unit quaternions.

Now we can define an action of the group of unit quaternions SU(2) on R3. For this, weuse the fact that R3 can be identified with the pure quaternions in H, namely, the quaternionsof the form x1i + x2j + x3k, where (x1, x2, x3) ∈ R3. Then, we define the action of SU(2)over R3 by

Z ·X = ZXZ−1 = ZXZ,

where Z ∈ SU(2) and X is any pure quaternion. Now, it turns out that the map ρZ (whereρZ(X) = ZXZ) is indeed a rotation, and that the map ρ : Z 7→ ρZ is a surjective homomor-phism ρ : SU(2) → SO(3) whose kernel is {−1, 1}, where 1 denotes the multiplicative unitquaternion. (For details, see Gallier [16], Chapter 8).

We can also define an action of the group SU(2)×SU(2) over R4, by identifying R4 withthe quaternions. In this case,

(Y, Z) ·X = Y XZ,

where (Y, Z) ∈ SU(2)×SU(2) andX ∈ H is any quaternion. Then, the map ρY,Z is a rotation

(where ρY,Z(X) = Y XZ), and the map ρ : (Y, Z) 7→ ρY,Z is a surjective homomorphismρ : SU(2)×SU(2) → SO(4) whose kernel is {(1, 1), (−1,−1)}. (For details, see Gallier [16],Chapter 8).

Thus, we observe that for n = 2, 3, 4, the rotations in SO(n) can be realized via thelinear action of some group (the case n = 1 is trivial, since SO(1) = {1,−1}). It is also thecase that the action of each group can be somehow be described in terms of multiplication insome larger algebra “containing” the original vector space Rn (C for n = 2, H for n = 3, 4).However, these groups appear to have been discovered in an ad hoc fashion, and there doesnot appear to be any universal way to define the action of these groups on Rn. It wouldcertainly be nice if the action was always of the form

Z ·X = ZXZ−1(= ZXZ).

A systematic way of constructing groups realizing rotations in terms of linear action, usinga uniform notion of action, does exist. Such groups are the spinor groups, to be describedin the following sections.

1.2 Clifford Algebras

We explained in Section 1.1 how the rotations in SO(3) can be realized by the linear action ofthe group of unit quaternions SU(2) on R3, and how the rotations in SO(4) can be realizedby the linear action of the group SU(2)× SU(2) on R4.

The main reasons why the rotations in SO(3) can be represented by unit quaternions arethe following:


(1) For every nonzero vector u ∈ R3, the reflection su about the hyperplane perpendicularto u is represented by the map

v 7→ −uvu−1,

where u and v are viewed as pure quaternions in H (i.e., if u = (u1, u2, u2), then viewu as u1i+ u2j+ u3k, and similarly for v).

(2) The group SO(3) is generated by the reflections.

As one can imagine, a successful generalization of the quaternions, i.e., the discoveryof a group G inducing the rotations in SO(n) via a linear action, depends on the abilityto generalize properties (1) and (2) above. Fortunately, it is true that the group SO(n) isgenerated by the hyperplane reflections. In fact, this is also true for the orthogonal groupO(n), and more generally for the group of direct isometries O(Φ) of any nondegeneratequadratic form Φ, by the Cartan-Dieudonne theorem (for instance, see Bourbaki [6], orGallier [16], Chapter 7, Theorem 7.2.1). In order to generalize (2), we need to understandhow the group G acts on Rn. Now, the case n = 3 is special, because the underlying spaceR3 on which the rotations act can be embedded as the pure quaternions in H. The casen = 4 is also special, because R

4 is the underlying space of H. The generalization to n ≥ 5requires more machinery, namely, the notions of Clifford groups and Clifford algebras.

As we will see, for every n ≥ 2, there is a compact, connected (and simply connected whenn ≥ 3) group Spin(n), the “spinor group,” and a surjective homomorphism ρ : Spin(n) →SO(n) whose kernel is {−1, 1}. This time, Spin(n) acts directly on R

n, because Spin(n) isa certain subgroup of the group of units of the Clifford algebra Cln, and Rn is naturally asubspace of Cln.

The group of unit quaternions SU(2) turns out to be isomorphic to the spinor groupSpin(3). Because Spin(3) acts directly on R3, the representation of rotations in SO(3)by elements of Spin(3) may be viewed as more natural than the representation by unitquaternions. The group SU(2) × SU(2) turns out to be isomorphic to the spinor groupSpin(4), but this isomorphism is less obvious.

In summary, we are going to define a group Spin(n) representing the rotations in SO(n),for any n ≥ 1, in the sense that there is a linear action of Spin(n) on Rn which induces asurjective homomorphism ρ : Spin(n) → SO(n) whose kernel is {−1, 1}. Furthermore, theaction of Spin(n) on Rn is given in terms of multiplication in an algebra Cln containingSpin(n), and in which Rn is also embedded.

It turns out that as a bonus, for n ≥ 3, the group Spin(n) is topologically simplerthan SO(n), since Spin(n) is simply connected, but SO(n) is not. By being astute, wecan also construct a group Pin(n) and a linear action of Pin(n) on Rn that induces asurjective homomorphism ρ : Pin(n) → O(n) whose kernel is {−1, 1}. The difficulty hereis the presence of the negative sign in (2). We will see how Atiyah, Bott and Shapirocircumvent this problem by using a “twisted adjoint action,” as opposed to the usual adjointaction (where v 7→ uvu−1).


Our presentation is heavily influenced by Brocker and tom Dieck [7] (Chapter 1, Section6), where most details can be found. This Chapter is almost entirely taken from the first 11pages of the beautiful and seminal paper by Atiyah, Bott and Shapiro [3], Clifford Modules,and we highly recommend it. Another excellent (but concise) exposition can be found inKirillov [18]. A very thorough exposition can be found in two places:

1. Lawson and Michelsohn [20], where the material on Pin(p, q) and Spin(p, q) can befound in Chapter I.

2. Lounesto’s excellent book [21].

One may also want to consult Baker [4], Curtis [12], Porteous [24], Fulton and Harris (Lecture20) [15], Choquet-Bruhat [11], Bourbaki [6], and Chevalley [10], a classic. The original sourceis Elie Cartan’s book (1937) whose translation in English appears in [8].

We begin by recalling what is an algebra over a field. Let K denote any (commutative)field, although for our purposes we may assume that K = R (and occasionally, K = C).Since we will only be dealing with associative algebras with a multiplicative unit, we onlydefine algebras of this kind.

Definition 1.1. Given a field K, a K-algebra is a K-vector space A together with a bilinearoperation · : A × A → A, called multiplication, which makes A into a ring with unity 1(or 1A, when we want to be very precise). This means that · is associative and that thereis a multiplicative identity element 1 so that 1 · a = a · 1 = a, for all a ∈ A. Given twoK-algebras A and B, a K-algebra homomorphism h : A → B is a linear map that is also aring homomorphism, with h(1A) = 1B.

For example, the ring Mn(K) of all n× n matrices over a field K is a K-algebra.

There is an obvious notion of ideal of a K-algebra: An ideal A ⊆ A is a linear subspaceof A that is also a two-sided ideal with respect to multiplication in A. If the field K isunderstood, we usually simply say an algebra instead of a K-algebra.

We will also need a quick review of tensor products. The basic idea is that tensor productsallow us to view multilinear maps as linear maps. The maps become simpler, but the spaces(product spaces) become more complicated (tensor products). For more details, see Atiyahand Macdonald [2].

Definition 1.2. Given two K-vector spaces E and F , a tensor product of E and F is a pair(E ⊗ F, ⊗), where E ⊗ F is a K-vector space and ⊗ : E × F → E ⊗ F is a bilinear map, sothat for every K-vector space G and every bilinear map f : E × F → G, there is a uniquelinear map f⊗ : E ⊗ F → G with

f(u, v) = f⊗(u⊗ v) for all u ∈ E and all v ∈ V ,


as in the diagram below:

E × F⊗

//

f%%▲

▲

▲

▲

▲

▲

▲

▲

▲

▲

▲

E ⊗ F

f⊗��

G

The vector space E ⊗ F is defined up to isomorphism. The vectors u⊗ v, where u ∈ Eand v ∈ F , generate E ⊗ F .

Remark: We should really denote the tensor product of E and F by E ⊗K F , since itdepends on the field K. Since we usually deal with a fixed field K, we use the simplernotation E ⊗ F .

We have natural isomorphisms

(E ⊗ F )⊗G ≈ E ⊗ (F ⊗G) and E ⊗ F ≈ F ⊗ E.

Given two linear maps f : E → F and g : E ′ → F ′, we have a unique bilinear mapf × g : E × E ′ → F × F ′ so that

(f × g)(a, a′) = (f(a), g(a′)) for all a ∈ E and all a′ ∈ E ′.

Thus, we have the bilinear map ⊗ ◦ (f × g) : E × E ′ → F ⊗ F ′, and so, there is a uniquelinear map f ⊗ g : E ⊗ E ′ → F ⊗ F ′ so that

(f ⊗ g)(a⊗ a′) = f(a)⊗ g(a′) for all a ∈ E and all a′ ∈ E ′.

Let us now assume that E and F are K-algebras. We want to make E ⊗ F into a K-algebra. Since the multiplication operations mE : E × E → E and mF : F × F → F arebilinear, we get linear maps m′

E : E⊗E → E and m′F : F ⊗F → F , and thus the linear map

m′E ⊗m′

F : (E ⊗ E)⊗ (F ⊗ F ) → E ⊗ F.

Using the isomorphism τ : (E ⊗ E)⊗ (F ⊗ F ) → (E ⊗ F )⊗ (E ⊗ F ), we get a linear map

mE⊗F : (E ⊗ F )⊗ (E ⊗ F ) → E ⊗ F,

which defines a multiplication m on E ⊗ F (namely, m(u, v) = mE⊗F (u ⊗ v)). It is easilychecked that E ⊗ F is indeed a K-algebra under the multiplication m. Using the simplernotation · for m, we have

(a⊗ a′) · (b⊗ b′) = (ab)⊗ (a′b′)

for all a, b ∈ E and all a′, b′ ∈ F .

Given any vector space V over a field K, there is a special K-algebra T (V ) togetherwith a linear map i : V → T (V ), with the following universal mapping property: Given any


K-algebra A, for any linear map f : V → A, there is a unique K-algebra homomorphismf : T (V ) → A so that

f = f ◦ i,as in the diagram below:

Vi//

f""❊

❊

❊

❊

❊

❊

❊

❊

❊

T (V )

f��

A

The algebra T (V ) is the tensor algebra of V . The algebra T (V ) may be constructed as thedirect sum

T (V ) =⊕

i≥0

V ⊗i,

where V 0 = K, and V ⊗i is the i-fold tensor product of V with itself. For every i ≥ 0, thereis a natural injection ιn : V

⊗n → T (V ), and in particular, an injection ι0 : K → T (V ). Themultiplicative unit 1 of T (V ) is the image ι0(1) in T (V ) of the unit 1 of the field K. Sinceevery v ∈ T (V ) can be expressed as a finite sum

v = v1 + · · ·+ vk,

where vi ∈ V ⊗ni and the ni are natural numbers with ni 6= nj if i 6= j, to define multiplicationin T (V ), using bilinearity, it is enough to define the multiplication V ⊗m×V ⊗n −→ V ⊗(m+n).Of course, this is defined by

(v1 ⊗ · · · ⊗ vm) · (w1 ⊗ · · · ⊗ wn) = v1 ⊗ · · · ⊗ vm ⊗ w1 ⊗ · · · ⊗ wn.

(This has to be made rigorous by using isomorphisms involving the associativity of tensorproducts; for details, see see Atiyah and Macdonald [2].) The algebra T (V ) is an exampleof a graded algebra, where the homogeneous elements of rank n are the elements in V ⊗n.

Remark: It is important to note that multiplication in T (V ) is not commutative. Also, inall rigor, the unit 1 of T (V ) is not equal to 1, the unit of the field K. However, in viewof the injection ι0 : K → T (V ), for the sake of notational simplicity, we will denote 1 by 1.More generally, in view of the injections ιn : V

⊗n → T (V ), we identify elements of V ⊗n withtheir images in T (V ).

Most algebras of interest arise as well-chosen quotients of the tensor algebra T (V ). Thisis true for the exterior algebra

∧• V (also called Grassmann algebra), where we take thequotient of T (V ) modulo the ideal generated by all elements of the form v⊗ v, where v ∈ V ,and for the symmetric algebra Sym V , where we take the quotient of T (V ) modulo the idealgenerated by all elements of the form v ⊗ w − w ⊗ v, where v, w ∈ V .

A Clifford algebra may be viewed as a refinement of the exterior algebra, in which we takethe quotient of T (V ) modulo the ideal generated by all elements of the form v⊗ v−Φ(v) · 1,where Φ is the quadratic form associated with a symmetric bilinear form ϕ : V × V → K,


and · : K × T (V ) → T (V ) denotes the scalar product of the algebra T (V ). For simplicity,let us assume that we are now dealing with real algebras.

Definition 1.3. Let V be a real finite-dimensional vector space together with a symmetricbilinear form ϕ : V × V → R and associated quadratic form Φ(v) = ϕ(v, v). A Cliffordalgebra associated with V and Φ is a real algebra Cl(V,Φ) together with a linear map iΦ : V →Cl(V,Φ) satisfying the condition (iΦ(v))

2 = Φ(v) · 1 for all v ∈ V , and so that for every realalgebra A and every linear map f : V → A with

(f(v))2 = Φ(v) · 1 for all v ∈ V ,

there is a unique algebra homomorphism f : Cl(V,Φ) → A so that

f = f ◦ iΦ,

as in the diagram below:

ViΦ//

f$$■

■

■

■

■

■

■

■

■

■

Cl(V,Φ)

f��

A

We use the notation λ ·u for the product of a scalar λ ∈ R and of an element u in the algebraCl(V,Φ), and juxtaposition uv for the multiplication of two elements u and v in the algebraCl(V,Φ).

By a familiar argument, any two Clifford algebras associated with V and Φ are isomorphic.We often denote iΦ by i.

To show the existence of Cl(V,Φ), observe that T (V )/A does the job, where A is theideal of T (V ) generated by all elements of the form v⊗ v−Φ(v) · 1, where v ∈ V . The mapiΦ : V → Cl(V,Φ) is the composition

Vι1−→ T (V )

π−→ T (V )/A,

where π is the natural quotient map. We often denote the Clifford algebra Cl(V,Φ) simplyby Cl(Φ).

Remark: Observe that Definition 1.3 does not assert that iΦ is injective or that there isan injection of R into Cl(V,Φ), but we will prove later that both facts are true when V isfinite-dimensional. Also, as in the case of the tensor algebra, the unit of the algebra Cl(V,Φ)and the unit of the field R are not equal.

SinceΦ(u+ v)− Φ(u)− Φ(v) = 2ϕ(u, v)

and(i(u+ v))2 = (i(u))2 + (i(v))2 + i(u)i(v) + i(v)i(u),


using the fact thati(u)2 = Φ(u) · 1,

we geti(u)i(v) + i(v)i(u) = 2ϕ(u, v) · 1.

As a consequence, if (u1, . . . , un) is an orthogonal basis w.r.t. ϕ (which means thatϕ(uj, uk) = 0 for all j 6= k), we have

i(uj)i(uk) + i(uk)i(uj) = 0 for all j 6= k.

Remark: Certain authors drop the unit 1 of the Clifford algebra Cl(V,Φ) when writing theidentities

i(u)2 = Φ(u) · 1and

2ϕ(u, v) · 1 = i(u)i(v) + i(v)i(u),

where the second identity is often written as

ϕ(u, v) =1

2(i(u)i(v) + i(v)i(u)).

This is very confusing and technically wrong, because we only have an injection of R intoCl(V,Φ), but R is not a subset of Cl(V,Φ).

� We warn the readers that Lawson and Michelsohn [20] adopt the opposite of our signconvention in defining Clifford algebras, i.e., they use the condition

(f(v))2 = −Φ(v) · 1 for all v ∈ V .

The most confusing consequence of this is that their Cl(p, q) is our Cl(q, p).

Observe that when Φ ≡ 0 is the quadratic form identically zero everywhere, then theClifford algebra Cl(V, 0) is just the exterior algebra

∧• V .

Example 1.1. Let V = R, e1 = 1, and assume that Φ(x1e1) = −x21. Then, Cl(Φ) is spanned

by the basis (1, e1). We havee21 = −1.

Under the bijectione1 7→ i,

the Clifford algebra Cl(Φ), also denoted by Cl1, is isomorphic to the algebra of complexnumbers C.

Now, let V = R2, (e1, e2) be the canonical basis, and assume that Φ(x1e1 + x2e2) =−(x2

1 + x22). Then, Cl(Φ) is spanned by the basis (1, e1, e2, e1e2). Furthermore, we have

e2e1 = −e1e2, e21 = −1, e22 = −1, (e1e2)2 = −1.


Under the bijectione1 7→ i, e2 7→ j, e1e2 7→ k,

it is easily checked that the quaternion identities

i2 = j2 = k2 = −1,

ij = −ji = k,

jk = −kj = i,

ki = −ik = j,

hold, and thus the Clifford algebra Cl(Φ), also denoted by Cl2, is isomorphic to the algebraof quaternions H.

Our prime goal is to define an action of Cl(Φ) on V in such a way that by restrictingthis action to some suitably chosen multiplicative subgroups of Cl(Φ), we get surjectivehomomorphisms onto O(Φ) and SO(Φ), respectively. The key point is that a reflection inV about a hyperplane H orthogonal to a vector w can be defined by such an action, butsome negative sign shows up. A correct handling of signs is a bit subtle and requires theintroduction of a canonical anti-automorphism t, and of a canonical automorphism α, definedas follows:

Proposition 1.1. Every Clifford algebra Cl(Φ) possesses a canonical anti-automorphismt : Cl(Φ) → Cl(Φ) satisfying the properties

t(xy) = t(y)t(x), t ◦ t = id, and t(i(v)) = i(v),

for all x, y ∈ Cl(Φ) and all v ∈ V . Furthermore, such an anti-automorphism is unique.

Proof. Consider the opposite algebra Cl(Φ)o, in which the product of x and y is given byyx. It has the universal mapping property. Thus, we get a unique isomorphism t, as in thediagram below:

Vi//

i##●

●

●

●

●

●

●

●

●

●

Cl(V,Φ)

t

��

Cl(Φ)o

We also denote t(x) by xt. When V is finite-dimensional, for a more palatable descriptionof t in terms of a basis of V , see the paragraph following Theorem 1.4.

The canonical automorphism α is defined using the proposition

Proposition 1.2. Every Clifford algebra Cl(Φ) has a unique canonical automorphismα : Cl(Φ) → Cl(Φ) satisfying the properties

α ◦ α = id, and α(i(v)) = −i(v),

for all v ∈ V .


Proof. Consider the linear map α0 : V → Cl(Φ) defined by α0(v) = −i(v), for all v ∈ V . Weget a unique homomorphism α as in the diagram below:

Vi//

α0##●

●

●

●

●

●

●

●

●

●

Cl(V,Φ)

α

��

Cl(Φ)

Furthermore, every x ∈ Cl(Φ) can be written as

x = x1 · · ·xm,

with xj ∈ i(V ), and since α(xj) = −xj , we get α ◦ α = id. It is clear that α is bijective.

Again, when V is finite-dimensional, a more palatable description of α in terms of a basisof V can be given. If (e1, . . . , en) is a basis of V , then the Clifford algebra Cl(Φ) consists ofcertain kinds of “polynomials,” linear combinations of monomials of the form

∑J λJeJ , where

J = {i1, i2, . . . , ik} is any subset (possibly empty) of {1, . . . , n}, with 1 ≤ i1 < i2 · · · < ik ≤ n,and the monomial eJ is the “product” ei1ei2 · · · eik . The map α is the linear map defined onmonomials by

α(ei1ei2 · · · eik) = (−1)kei1ei2 · · · eik .For a more rigorous explanation, see the paragraph following Theorem 1.4.

We now show that if V has dimension n, then i is injective and Cl(Φ) has dimension 2n.A clever way of doing this is to introduce a graded tensor product.

First, observe thatCl(Φ) = Cl0(Φ)⊕ Cl1(Φ),

whereCli(Φ) = {x ∈ Cl(Φ) | α(x) = (−1)ix}, where i = 0, 1.

We say that we have a Z/2-grading , which means that if x ∈ Cli(Φ) and y ∈ Clj(Φ), thenxy ∈ Cli+j (mod 2)(Φ).

When V is finite-dimensional, since every element of Cl(Φ) is a linear combination of theform

∑J λJeJ as explained earlier, in view of the description of α given above, we see that

the elements of Cl0(Φ) are those for which the monomials eJ are products of an even numberof factors, and the elements of Cl1(Φ) are those for which the monomials eJ are products ofan odd number of factors.

Remark: Observe that Cl0(Φ) is a subalgebra of Cl(Φ), whereas Cl1(Φ) is not.

Given two Z/2-graded algebras A = A0 ⊕ A1 and B = B0 ⊕ B1, their graded tensorproduct A ⊗ B is defined by

(A ⊗ B)0 = (A0 ⊗B0)⊕ (A1 ⊗ B1),

(A ⊗ B)1 = (A0 ⊗B1)⊕ (A1 ⊗ B0),


with multiplication(a′ ⊗ b)(a⊗ b′) = (−1)ij(a′a)⊗ (bb′),

for a ∈ Ai and b ∈ Bj. The reader should check that A ⊗ B is indeed Z/2-graded.

Proposition 1.3. Let V and W be finite dimensional vector spaces with quadratic forms Φand Ψ. Then, there is a quadratic form Φ⊕Ψ on V ⊕W defined by

(Φ + Ψ)(v, w) = Φ(v) + Ψ(w).

If we write i : V → Cl(Φ) and j : W → Cl(Ψ), we can define a linear map

f : V ⊕W → Cl(Φ) ⊗ Cl(Ψ)

byf(v, w) = i(v)⊗ 1 + 1⊗ j(w).

Furthermore, the map f induces an isomorphism (also denoted by f)

f : Cl(V ⊕W ) → Cl(Φ) ⊗ Cl(Ψ).

Proof. See Brocker and tom Dieck [7], Chapter 1, Section 6, page 57.

As a corollary, we obtain the following result:

Theorem 1.4. For every vector space V of finite dimension n, the map i : V → Cl(Φ) isinjective. Given a basis (e1, . . . , en) of V , the 2n − 1 products

i(ei1)i(ei2) · · · i(eik), 1 ≤ i1 < i2 · · · < ik ≤ n,

and 1 form a basis of Cl(Φ). Thus, Cl(Φ) has dimension 2n.

Proof. The proof is by induction on n = dim(V ). For n = 1, the tensor algebra T (V ) is justthe polynomial ring R[X ], where i(e1) = X . Thus, Cl(Φ) = R[X ]/(X2 − Φ(e1)), and theresult is obvious. Since

i(ej)i(ek) + i(ek)i(ej) = 2ϕ(ei, ej) · 1,

it is clear that the products

i(ei1)i(ei2) · · · i(eik), 1 ≤ i1 < i2 · · · < ik ≤ n,

and 1 generate Cl(Φ). Now, there is always a basis that is orthogonal with respect to ϕ (forexample, see Artin [1], Chapter 7, or Gallier [16], Chapter 6, Problem 6.14), and thus, wehave a splitting

(V,Φ) =

n⊕

k=1

(Vk,Φk),

where Vk has dimension 1. Choosing a basis so that ek ∈ Vk, the theorem follows by inductionfrom Proposition 1.3.

1.3. CLIFFORD GROUPS 19

Since i is injective, for simplicity of notation, from now on we write u for i(u). Theorem1.4 implies that if (e1, . . . , en) is an orthogonal basis of V , then Cl(Φ) is the algebra presentedby the generators (e1, . . . , en) and the relations

e2j = Φ(ej) · 1, 1 ≤ j ≤ n, and

ejek = −ekej , 1 ≤ j, k ≤ n, j 6= k.

If V has finite dimension n and (e1, . . . , en) is a basis of V , by Theorem 1.4, the maps tand α are completely determined by their action on the basis elements. Namely, t is definedby

t(ei) = ei

t(ei1ei2 · · · eik) = eikeik−1· · · ei1 ,

where 1 ≤ i1 < i2 · · · < ik ≤ n, and of course, t(1) = 1. The map α is defined by

α(ei) = −ei

α(ei1ei2 · · · eik) = (−1)kei1ei2 · · · eik

where 1 ≤ i1 < i2 · · · < ik ≤ n, and of course, α(1) = 1. Furthermore, the even-gradedelements (the elements of Cl0(Φ)) are those generated by 1 and the basis elements consistingof an even number of factors ei1ei2 · · · ei2k , and the odd-graded elements (the elements ofCl1(Φ)) are those generated by the basis elements consisting of an odd number of factorsei1ei2 · · · ei2k+1

.

We are now ready to define the Clifford group and investigate some of its properties.

1.3 Clifford Groups

First, we define conjugation on a Clifford algebra Cl(Φ) as the map

x 7→ x = t(α(x)) for all x ∈ Cl(Φ).

Observe thatt ◦ α = α ◦ t.

If V has finite dimension n and (e1, . . . , en) is a basis of V , in view of previous remarks,conjugation is defined by

ei = −ei

ei1ei2 · · · eik = (−1)keikeik−1· · · ei1

where 1 ≤ i1 < i2 · · · < ik ≤ n, and of course, 1 = 1. Conjugation is an anti-automorphism.


The multiplicative group of invertible elements of Cl(Φ) is denoted by Cl(Φ)∗. Observethat for any x ∈ V , if Φ(x) 6= 0, then x is invertible because x2 = Φ(x); that is, x ∈ Cl(Φ)∗.

We would like Cl(Φ)∗ to act on V via

x · v = α(x)vx−1,

where x ∈ Cl(Φ)∗ and v ∈ V . In general, there is no reason why α(x)vx−1 should be in V orwhy this action defines an automorphism of V , so we restrict this map to the subset Γ(Φ)of Cl(Φ)∗ as follows.

Definition 1.4. Given a finite dimensional vector space V and a quadratic form Φ on V ,the Clifford group of Φ is the group

Γ(Φ) = {x ∈ Cl(Φ)∗ | α(x)vx−1 ∈ V for all v ∈ V }.

The map N : Cl(Q) → Cl(Q) given by

N(x) = xx

is called the norm of Cl(Φ).

For any x ∈ Γ(Φ), let ρx : V → V be the map defined by

v 7→ α(x)vx−1, v ∈ V.

It is not entirely obvious why the map ρ : Γ(Φ) → GL(V ) given by x 7→ ρx is a linear action,and for that matter, why Γ(Φ) is a group. This is because V is finite-dimensional and α isan automorphism.

Proof. For any x ∈ Γ(Φ), the map ρx from V to V defined by

v 7→ α(x)vx−1

is clearly linear. If α(x)vx−1 = 0, since by hypothesis x is invertible and since α is anautomorphism α(x) is also invertible, so v = 0. Thus our linear map is injective, and sinceV has finite dimension, it is bijective. To prove that x−1 ∈ Γ(Φ), pick any v ∈ V . Sincethe linear map ρx is bijective, there is some w ∈ V such that ρx(w) = v, which means thatα(x)wx−1 = v. Since x is invertible and α is an automorphism, we get

α(x−1)vx = w,

so α(x−1)vx ∈ V ; since this holds for any v ∈ V , we have x−1 ∈ Γ(Φ). Since α is anautomorphism, if x, y ∈ Γ(Φ), for any v ∈ V we have

ρy(ρx(v)) = α(y)α(x)vx−1y−1 = α(yx)v(yx)−1 = ρyx(v),

which shows that ρyx is a linear automorphism of V , so yx ∈ Γ(Φ) and ρ is a homomorphism.Therefore, Γ(Φ) is a group and ρ is a linear representation.


We also define the group Γ+(Φ), called the special Clifford group, by

Γ+(Φ) = Γ(Φ) ∩ Cl0(Φ).

Observe that N(v) = −Φ(v) · 1 for all v ∈ V . Also, if (e1, . . . , en) is a basis of V , we leave itas an exercise to check that

N(ei1ei2 · · · eik) = (−1)kΦ(ei1)Φ(ei2) · · ·Φ(eik) · 1.

Remark: The map ρ : Γ(Φ) → GL(V ) given by x 7→ ρx is called the twisted adjoint rep-resentation. It was introduced by Atiyah, Bott and Shapiro [3]. It has the advantage ofnot introducing a spurious negative sign, i.e., when v ∈ V and Φ(v) 6= 0, the map ρv is thereflection sv about the hyperplane orthogonal to v (see Proposition 1.6). Furthermore, whenΦ is nondegenerate, the kernel Ker (ρ) of the representation ρ is given by Ker (ρ) = R∗ · 1,where R

∗ = R− {0}. The earlier adjoint representation (used by Chevalley [10] and others)is given by

v 7→ xvx−1.

Unfortunately, in this case, ρx represents −sv, where sv is the reflection about the hyperplaneorthogonal to v. Furthermore, the kernel of the representation ρ is generally bigger than R∗·1.This is the reason why the twisted adjoint representation is preferred (and must be used fora proper treatment of the Pin group).

Proposition 1.5. The maps α and t induce an automorphism and an anti-automorphismof the Clifford group, Γ(Φ).

Proof. It is not very instructive; see Brocker and tom Dieck [7], Chapter 1, Section 6, page58.

The following proposition shows why Clifford groups generalize the quaternions.

Proposition 1.6. Let V be a finite dimensional vector space and Φ a quadratic form onV . For every element x of the Clifford group Γ(Φ), if x ∈ V and Φ(x) 6= 0, then the mapρx : V → V given by

v 7→ α(x)vx−1 for all v ∈ V

is the reflection about the hyperplane H orthogonal to the vector x.

Proof. Recall that the reflection s about the hyperplane H orthogonal to the vector x isgiven by

s(u) = u− 2ϕ(u, x)

Φ(x)· x.

However, we havex2 = Φ(x) · 1 and ux+ xu = 2ϕ(u, x) · 1.


Thus, we have

s(u) = u− 2ϕ(u, x)

Φ(x)· x

= u− 2ϕ(u, x) ·(

1

Φ(x)· x

)

= u− 2ϕ(u, x) · x−1

= u− 2ϕ(u, x) · (1x−1)

= u− (2ϕ(u, x) · 1)x−1

= u− (ux+ xu)x−1

= −xux−1

= α(x)ux−1,

since α(x) = −x, for x ∈ V .

Recall that the linear representation

ρ : Γ(Φ) → GL(V )

is given byρ(x)(v) = α(x)vx−1,

for all x ∈ Γ(Φ) and all v ∈ V . We would like to show that ρ is a surjective homomorphismfrom Γ(Φ) onto O(ϕ), and a surjective homomorphism from Γ+(Φ) onto SO(ϕ). For this,we will need to assume that ϕ is nondegenerate, which means that for every v ∈ V , ifϕ(v, w) = 0 for all w ∈ V , then v = 0. For simplicity of exposition, we first assume that Φis the quadratic form on Rn defined by

Φ(x1, . . . , xn) = −(x21 + · · ·+ x2

n).

Let Cln denote the Clifford algebra Cl(Φ) and Γn denote the Clifford group Γ(Φ). Thefollowing lemma plays a crucial role:

Lemma 1.7. The kernel of the map ρ : Γn → GL(n) is R∗ · 1, the multiplicative group ofnonzero scalar multiples of 1 ∈ Cln.

Proof. If ρ(x) = id, thenα(x)v = vx for all v ∈ Rn. (1)

Since Cln = Cl0n ⊕ Cl1n, we can write x = x0 + x1, with xi ∈ Clin for i = 0, 1. Then, equation(1) becomes

x0v = vx0 and − x1v = vx1 for all v ∈ Rn. (2)

Using Theorem 1.4, we can express x0 as a linear combination of monomials in the canonicalbasis (e1, . . . , en), so that

x0 = a0 + e1b1, with a0 ∈ Cl0n, b

1 ∈ Cl1n,


where neither a0 nor b1 contains a summand with a factor e1. Applying the first relation in(2) to v = e1, we get

e1a0 + e21b

1 = a0e1 + e1b1e1. (3)

Now, the basis (e1, . . . , en) is orthogonal w.r.t. Φ, which implies that

ejek = −ekej for all j 6= k.

Since each monomial in a0 is of even degree and contains no factor e1, we get

a0e1 = e1a0.

Similarly, since b1 is of odd degree and contains no factor e1, we get

e1b1e1 = −e21b

1.

But then, from (3), we get

e1a0 + e21b

1 = a0e1 + e1b1e1 = e1a

0 − e21b1,

and so, e21b1 = 0. However, e21 = −1, and so, b1 = 0. Therefore, x0 contains no monomial

with a factor e1. We can apply the same argument to the other basis elements e2, . . . , en,and thus, we just proved that x0 ∈ R · 1.

A similar argument applying to the second equation in (2), with x1 = a1+e1b0 and v = e1

shows that b0 = 0. We also conclude that x1 ∈ R · 1. However, R · 1 ⊆ Cl0n, and so x1 = 0.Finally, x = x0 ∈ (R · 1) ∩ Γn = R∗ · 1.

Remark: If Φ is any nondegenerate quadratic form, we know (for instance, see Artin [1],Chapter 7, or Gallier [16], Chapter 6, Problem 6.14) that there is an orthogonal basis(e1, . . . , en) with respect to ϕ (i.e. ϕ(ej , ek) = 0 for all j 6= k). Thus, the commutationrelations

e2j = Φ(ej) · 1, with Φ(ej) 6= 0, 1 ≤ j ≤ n, and

ejek = −ekej , 1 ≤ j, k ≤ n, j 6= k

hold, and since the proof only rests on these facts, Lemma 1.7 holds for any nondegeneratequadratic form.

� However, Lemma 1.7 may fail for degenerate quadratic forms. For example, if Φ ≡ 0,then Cl(V, 0) =

∧• V . Consider the element x = 1 + e1e2. Clearly, x−1 = 1− e1e2. But

now, for any v ∈ V , we have

α(1 + e1e2)v(1 + e1e2)−1 = (1 + e1e2)v(1− e1e2) = v.

Yet, 1 + e1e2 is not a scalar multiple of 1.


The following proposition shows that the notion of norm is well-behaved.

Proposition 1.8. If x ∈ Γn, then N(x) ∈ R∗ · 1.Proof. The trick is to show that N(x) is in the kernel of ρ. To say that x ∈ Γn means that

α(x)vx−1 ∈ Rn for all v ∈ Rn.

Applying t, we gett(x)−1vt(α(x)) = α(x)vx−1,

since t is the identity on Rn. Thus, we have

v = t(x)α(x)v(t(α(x))x)−1 = α(xx)v(xx)−1,

so xx ∈ Ker (ρ). By Proposition 1.5, we have x ∈ Γn, and so, xx = xx ∈ Ker (ρ).

Remark: Again, the proof also holds for the Clifford group Γ(Φ) associated with any non-degenerate quadratic form Φ. When Φ(v) = −‖v‖2, where ‖v‖ is the standard Euclideannorm of v, we have N(v) = ‖v‖2 · 1 for all v ∈ V . However, for other quadratic forms, it ispossible that N(x) = λ · 1 where λ < 0, and this is a difficulty that needs to be overcome.

Proposition 1.9. The restriction of the norm N to Γn is a homomorphism N : Γn → R∗ ·1,and N(α(x)) = N(x) for all x ∈ Γn.

Proof. We haveN(xy) = xyy x = xN(y)x = xxN(y) = N(x)N(y),

where the third equality holds because N(x) ∈ R∗ · 1. We also have

N(α(x)) = α(x)α(x) = α(xx) = α(N(x)) = N(x).

Remark: The proof also holds for the Clifford group Γ(Φ) associated with any nondegen-erate quadratic form Φ.

Proposition 1.10. We have Rn − {0} ⊆ Γn and ρ(Γn) ⊆ O(n).

Proof. Let x ∈ Γn and v ∈ Rn, with v 6= 0. We have

N(ρ(x)(v)) = N(α(x)vx−1) = N(α(x))N(v)N(x−1) = N(x)N(v)N(x)−1 = N(v),

since N : Γn → R∗ · 1. However, for v ∈ Rn, we know that

N(v) = −Φ(v) · 1.Thus, ρ(x) is norm-preserving, and so, ρ(x) ∈ O(n).

Remark: The proof that ρ(Γ(Φ)) ⊆ O(Φ) also holds for the Clifford group Γ(Φ) associatedwith any nondegenerate quadratic form Φ. The first statement needs to be replaced by thefact that every non-isotropic vector in R

n (a vector is non-isotropic if Φ(x) 6= 0) belongs toΓ(Φ). Indeed, x2 = Φ(x) · 1, which implies that x is invertible.

We are finally ready for the introduction of the groups Pin(n) and Spin(n).

1.4. THE GROUPS PIN(N) AND SPIN(N) 25

1.4 The Groups Pin(n) and Spin(n)

Definition 1.5. We define the pinor group Pin(n) as the kernel Ker (N) of the homomor-phism N : Γn → R∗ · 1, and the spinor group Spin(n) as Pin(n) ∩ Γ+

n .

Observe that if N(x) = 1, then x is invertible, and x−1 = x since xx = N(x) = 1. Thus,we can write

Pin(n) = {x ∈ Cln | α(x)vx−1 ∈ Rn for all v ∈ Rn, N(x) = 1}

= {x ∈ Cln | α(x)vx ∈ Rn for all v ∈ R

n, xx = 1},

and

Spin(n) = {x ∈ Cl0n | xvx−1 ∈ Rn for all v ∈ Rn, N(x) = 1}

= {x ∈ Cl0n | xvx ∈ Rn for all v ∈ Rn, xx = 1}

Remark: According to Atiyah, Bott and Shapiro, the use of the name Pin(k) is a joke dueto Jean-Pierre Serre (Atiyah, Bott and Shapiro [3], page 1).

Theorem 1.11. The restriction of ρ : Γn → O(n) to the pinor group Pin(n) is a surjectivehomomorphism ρ : Pin(n) → O(n) whose kernel is {−1, 1}, and the restriction of ρ to thespinor group Spin(n) is a surjective homomorphism ρ : Spin(n) → SO(n) whose kernel is{−1, 1}.

Proof. By Proposition 1.10, we have a map ρ : Pin(n) → O(n). The reader can easily checkthat ρ is a homomorphism. By the Cartan-Dieudonne theorem (see Bourbaki [6], or Gallier[16], Chapter 7, Theorem 7.2.1), every isometry f ∈ SO(n) is the composition f = s1◦· · ·◦skof hyperplane reflections sj . If we assume that sj is a reflection about the hyperplane Hj

orthogonal to the nonzero vector wj , by Proposition 1.6, ρ(wj) = sj. Since N(wj) = ‖wj‖2 ·1,we can replace wj by wj/ ‖wj‖, so that N(w1 · · ·wk) = 1, and then

f = ρ(w1 · · ·wk),

and ρ is surjective. Note that

Ker (ρ | Pin(n)) = Ker (ρ) ∩ ker(N) = {t ∈ R∗ · 1 | N(t) = 1} = {−1, 1}.

As to Spin(n), we just need to show that the restriction of ρ to Spin(n) maps Γn intoSO(n). If this was not the case, there would be some improper isometry f ∈ O(n) so thatρ(x) = f , where x ∈ Γn ∩ Cl0n. However, we can express f as the composition of an oddnumber of reflections, say

f = ρ(w1 · · ·w2k+1).

Sinceρ(w1 · · ·w2k+1) = ρ(x),


we have x−1w1 · · ·w2k+1 ∈ Ker (ρ). By Lemma 1.7, we must have

x−1w1 · · ·w2k+1 = λ · 1

for some λ ∈ R∗, and thusw1 · · ·w2k+1 = λ · x,

where x has even degree and w1 · · ·w2k+1 has odd degree, which is impossible.

Let us denote the set of elements v ∈ Rn with N(v) = 1 (with norm 1) by Sn−1. We havethe following corollary of Theorem 1.11:

Corollary 1.12. The group Pin(n) is generated by Sn−1, and every element of Spin(n) canbe written as the product of an even number of elements of Sn−1.

Example 1.2. The reader should verify that

Pin(1) ≈ Z/4Z, Spin(1) = {−1, 1} ≈ Z/2Z,

and also that

Pin(2) ≈ {ae1 + be2 | a2 + b2 = 1} ∪ {c1 + de1e2 | c2 + d2 = 1}, Spin(2) = U(1).

We may also write Pin(2) = U(1) + U(1), where U(1) is the group of complex numbersof modulus 1 (the unit circle in R2). It can also be shown that Spin(3) ≈ SU(2) andSpin(4) ≈ SU(2) × SU(2). The group Spin(5) is isomorphic to the symplectic groupSp(2), and Spin(6) is isomorphic to SU(4) (see Curtis [12] or Porteous [24]).

Let us take a closer look at Spin(2). The Clifford algebra Cl2 is generated by the fourelements

1, e1, e2, , e1e2,

and they satisfy the relations

e21 = −1, e22 = −1, e1e2 = −e2e1.

The group Spin(2) consists of all products

2k∏

i=1

(aie1 + bie2)

consisting of an even number of factors and such that a2i + b2i = 1. In view of the aboverelations, every such element can be written as

x = a1 + be1e2,


where x satisfies the conditions that xvx−1 ∈ R2 for all v ∈ R2, and N(x) = 1. Since

X = a1− be1e2,

we getN(x) = a2 + b2,

and the condition N(x) = 1 is simply a2 + b2 = 1.

We claim that if x ∈ Cl02, then xvx−1 ∈ R2. Indeed, since x ∈ Cl02 and v ∈ Cl12, we have

xvx−1 ∈ Cl12, which implies that xvx−1 ∈ R2, since the only elements of Cl12 are those inR2. Then, Spin(2) consists of those elements x = a1 + be1e2 so that a2 + b2 = 1. If we leti = e1e2, we observe that

i2 = −1,

e1i = −ie1 = −e2,

e2i = −ie2 = e1.

Thus, Spin(2) is isomorphic to U(1). Also note that

e1(a1 + bi) = (a1− bi)e1.

Let us find out explicitly what is the action of Spin(2) on R2. Given X = a1 + bi, witha2 + b2 = 1, for any v = v1e1 + v2e2, we have

α(X)vX−1 = X(v1e1 + v2e2)X−1

= X(v1e1 + v2e2)(−e1e1)X

= X(v1e1 + v2e2)(−e1)(e1X)

= X(v11 + v2i)Xe1

= X2(v11 + v2i)e1

= (((a2 − b2)v1 − 2abv2)1 + (a2 − b2)v2 + 2abv1)i)e1

= ((a2 − b2)v1 − 2abv2)e1 + (a2 − b2)v2 + 2abv1)e2.

Since a2+b2 = 1, we can write X = a1+bi = (cos θ)1+(sin θ)i, and the above derivationshows that

α(X)vX−1 = (cos 2θv1 − sin 2θv2)e1 + (cos 2θv2 + sin 2θv1)e2.

This means that the rotation ρX induced by X ∈ Spin(2) is the rotation of angle 2θ aroundthe origin. Observe that the maps

v 7→ v(−e1), X 7→ Xe1

establish bijections between R2 and Spin(2) ≃ U(1). Also, note that the action of X =cos θ+ i sin θ viewed as a complex number yields the rotation of angle θ, whereas the action


of X = (cos θ)1 + (sin θ)i viewed as a member of Spin(2) yields the rotation of angle 2θ.There is nothing wrong. In general, Spin(n) is a two–to–one cover of SO(n).

Next, let us take a closer look at Spin(3). The Clifford algebra Cl3 is generated by theeight elements

1, e1, e2, , e3, , e1e2, e2e3, e3e1, e1e2e3,

and they satisfy the relations

e2i = −1, ejej = −ejei, 1 ≤ i, j ≤ 3, i 6= j.

The group Spin(3) consists of all products

2k∏

i=1

(aie1 + bie2 + cie3)

consisting of an even number of factors and such that a2i + b2i + c2i = 1. In view of the aboverelations, every such element can be written as

x = a1 + be2e3 + ce3e1 + de1e2,

where x satisfies the conditions that xvx−1 ∈ R3 for all v ∈ R3, and N(x) = 1. Since

X = a1− be2e3 − ce3e1 − de1e2,

we getN(x) = a2 + b2 + c2 + d2,

and the condition N(x) = 1 is simply a2 + b2 + c2 + d2 = 1.

It turns out that the conditions x ∈ Cl03 and N(x) = 1 imply that xvx−1 ∈ R3 for allv ∈ R3. To prove this, first observe that N(x) = 1 implies that x−1 = ±x, and that v = −vfor any v ∈ R3, and so,

xvx−1 = −xvx−1.

Also, since x ∈ Cl03 and v ∈ Cl13, we have xvx−1 ∈ Cl13. Thus, we can write

xvx−1 = u+ λe1e2e3, for some u ∈ R3 and some λ ∈ R.

Bute1e2e3 = −e3e2e1 = e1e2e3,

and so,xvx−1 = −u+ λe1e2e3 = −xvx−1 = −u− λe1e2e3,

which implies that λ = 0. Thus, xvx−1 ∈ R3, as claimed. Then, Spin(3) consists of thoseelements x = a1 + be2e3 + ce3e1 + de1e2 so that a2 + b2 + c2 + d2 = 1. Under the bijection

i 7→ e2e3, j 7→ e3e1, k 7→ e1e2,


we can check that we have an isomorphism between the group SU(2) of unit quaternionsand Spin(3). If X = a1 + be2e3 + ce3e1 + de1e2 ∈ Spin(3), observe that

X−1 = X = a1− be2e3 − ce3e1 − de1e2.

Now, using the identification

i 7→ e2e3, j 7→ e3e1, k 7→ e1e2,

we can easily check that

(e1e2e3)2 = 1,

(e1e2e3)i = i(e1e2e3) = −e1,

(e1e2e3)j = j(e1e2e3) = −e2,

(e1e2e3)k = k(e1e2e3) = −e3,

(e1e2e3)e1 = −i,

(e1e2e3)e2 = −j,

(e1e2e3)e3 = −k.

Then, if X = a1 + bi + cj + dk ∈ Spin(3), for every v = v1e1 + v2e2 + v3e3, we have

α(X)vX−1 = X(v1e1 + v2e2 + v3e3)X−1

= X(e1e2e3)2(v1e1 + v2e2 + v3e3)X

−1

= (e1e2e3)X(e1e2e3)(v1e1 + v2e2 + v3e3)X−1

= −(e1e2e3)X(v1i + v2j + v3k)X−1.

This shows that the rotation ρX ∈ SO(3) induced by X ∈ Spin(3) can be viewed as therotation induced by the quaternion a1+bi+cj+dk on the pure quaternions, using the maps

v 7→ −(e1e2e3)v, X 7→ −(e1e2e3)X

to go from a vector v = v1e1 + v2e2 + v3e3 to the pure quaternion v1i+ v2j+ v3k, and back.

We close this section by taking a closer look at Spin(4). The group Spin(4) consists ofall products

2k∏

i=1

(aie1 + bie2 + cie3 + die4)

consisting of an even number of factors and such that a2i + b2i + c2i + d2i = 1. Using therelations

e2i = −1, ejej = −ejei, 1 ≤ i, j ≤ 4, i 6= j,

every element of Spin(4) can be written as

x = a11 + a2e1e2 + a3e2e3 + a4e3e1 + a5e4e3 + a6e4e1 + a7e4e2 + a8e1e2e3e4,


where x satisfies the conditions that xvx−1 ∈ R4 for all v ∈ R4, and N(x) = 1. Let

i = e1e2, j = e2e3, k = 33e1, i′ = e4e3, j

′ = e4e1, k′ = e4e2,

and I = e1e2e3e4. The reader will easily verify that

ij = k

jk = i

ki = j

i2 = −1, j2 = −1, k2 = −1

iI = Ii = i′

jI = Ij = j′

kI = Ik = k′

I2 = 1, I = I.

Then, every x ∈ Spin(4) can be written as

x = u+ Iv, with u = a1 + bi + cj+ dk and v = a′1 + b′i+ c′j + d′k,

with the extra conditions stated above. Using the above identities, we have

(u+ Iv)(u′ + Iv′) = uu′ + vv′ + I(uv′ + vu′).

As a consequence,

N(u + Iv) = (u+ Iv)(u+ Iv) = uu+ vv + I(uv + vu),

and thus, N(u + Iv) = 1 is equivalent to

uu+ vv = 1 and uv + vu = 0.

As in the case n = 3, it turns out that the conditions x ∈ Cl04 and N(x) = 1 imply thatxvx−1 ∈ R

4 for all v ∈ R4. The only change to the proof is that xvx−1 ∈ Cl14 can be written

asxvx−1 = u+

∑

i,j,k

λi,j,keiejek, for some u ∈ R4, with {i, j, k} ⊆ {1, 2, 3, 4}.

As in the previous proof, we get λi,j,k = 0. Then, Spin(4) consists of those elements u+ Ivso that

uu+ vv = 1 and uv + vu = 0,

with u and v of the form a1 + bi + cj + dk. Finally, we see that Spin(4) is isomorphic toSpin(3)× Spin(3) under the isomorphism

u+ vI 7→ (u+ v, u− v).

1.5. THE GROUPS PIN(P,Q) AND SPIN(P,Q) 31

Indeed, we haveN(u+ v) = (u+ v)(u+ v) = 1,

andN(u− v) = (u− v)(u− v) = 1,

sinceuu+ vv = 1 and uv + vu = 0,

and

(u+ v, u− v)(u′ + v′, u′ − v′) = (uu′ + vv′ + uv′ + vu′, uu′ + vv′ − (uv′ + vu′)).

Remark: It can be shown that the assertion if x ∈ Cl0n and N(x) = 1, then xvx−1 ∈ Rn forall v ∈ Rn, is true up to n = 5 (see Porteous [24], Chapter 13, Proposition 13.58). However,this is already false for n = 6. For example, if X = 1/

√2(1 + e1e2e3e4e5e6), it is easy to see

that N(X) = 1, and yet, Xe1X−1 /∈ R6.

1.5 The Groups Pin(p, q) and Spin(p, q)

For every nondegenerate quadratic form Φ over R, there is an orthogonal basis with respectto which Φ is given by

Φ(x1, . . . , xp+q) = x21 + · · ·+ x2

p − (x2p+1 + · · ·+ x2

p+q),

where p and q only depend on Φ. The quadratic form corresponding to (p, q) is denoted Φp,q

and we call (p, q) the signature of Φp,q. Let n = p + q. We define the group O(p, q) as thegroup of isometries w.r.t. Φp,q, i.e., the group of linear maps f so that

Φp,q(f(v)) = Φp,q(v) for all v ∈ Rn

and the group SO(p, q) as the subgroup of O(p, q) consisting of the isometries f ∈ O(p, q)with det(f) = 1. We denote the Clifford algebra Cl(Φp,q) where Φp,q has signature (p, q) byClp,q, the corresponding Clifford group by Γp,q, and the special Clifford group Γp,q ∩Cl0p,q byΓ+p,q. Note that with this new notation, Cln = Cl0,n.

� As we mentioned earlier, since Lawson and Michelsohn [20] adopt the opposite of oursign convention in defining Clifford algebras; their Cl(p, q) is our Cl(q, p).

As we mentioned in Section 1.3, we have the problem that N(v) = −Φ(v) · 1, but −Φ(v)is not necessarily positive (where v ∈ Rn). The fix is simple: Allow elements x ∈ Γp,q withN(x) = ±1.

Definition 1.6. We define the pinor group Pin(p, q) as the group

Pin(p, q) = {x ∈ Γp,q | N(x) = ±1},

and the spinor group Spin(p, q) as Pin(p, q) ∩ Γ+p,q.


Remarks:

(1) It is easily checked that the group Spin(p, q) is also given by

Spin(p, q) = {x ∈ Cl0p,q | xvx ∈ Rn for all v ∈ Rn, N(x) = 1}.

This is because Spin(p, q) consists of elements of even degree.

(2) One can check that if N(x) 6= 0, then

α(x)vx−1 = xvt(x)/N(x).

Thus, we have

Pin(p, q) = {x ∈ Clp,q | xvt(x)N(x) ∈ Rn for all v ∈ Rn, N(x) = ±1}.

When Φ(x) = −‖x‖2, we have N(x) = ‖x‖2, and

Pin(n) = {x ∈ Cln | xvt(x) ∈ Rn for all v ∈ Rn, N(x) = 1}.

Theorem 1.11 generalizes as follows:

Theorem 1.13. The restriction of ρ : Γp,q → GL(n) to the pinor group Pin(p, q) is asurjective homomorphism ρ : Pin(p, q) → O(p, q) whose kernel is {−1, 1}, and the restrictionof ρ to the spinor group Spin(p, q) is a surjective homomorphism ρ : Spin(p, q) → SO(p, q)whose kernel is {−1, 1}.

Proof. The Cartan-Dieudonne also holds for any nondegenerate quadratic form Φ, in thesense that every isometry in O(Φ) is the composition of reflections defined by hyperplanesorthogonal to non-isotropic vectors (see Dieudonne [13], Chevalley [10], Bourbaki [6], orGallier [16], Chapter 7, Problem 7.14). Thus, Theorem 1.11 also holds for any nondegeneratequadratic form Φ. The only change to the proof is the following: Since N(wj) = −Φ(wj) · 1,we can replace wj by wj/

√|Φ(wj)|, so that N(w1 · · ·wk) = ±1, and then

f = ρ(w1 · · ·wk),

and ρ is surjective.

If we consider Rn equipped with the quadratic form Φp,q (with n = p+ q), we denote theset of elements v ∈ R

n with N(v) = 1 by Sn−1p,q . We have the following corollary of Theorem

1.13 (generalizing Corollary 1.14):

Corollary 1.14. The group Pin(p, q) is generated by Sn−1p,q , and every element of Spin(p, q)

can be written as the product of an even number of elements of Sn−1p,q .

1.6. PERIODICITY OF THE CLIFFORD ALGEBRAS CLP,Q 33

Example 1.3. The reader should check that

Cl0,1 ≈ C, Cl1,0 ≈ R⊕ R.

We also havePin(0, 1) ≈ Z/4Z, Pin(1, 0) ≈ Z/2Z× Z/2Z,

from which we get Spin(0, 1) = Spin(1, 0) ≈ Z/2Z. Also, show that

Cl0,2 ≈ H, Cl1,1 ≈ M2(R), Cl2,0 ≈ M2(R),

where Mn(R) denotes the algebra of n × n matrices. One can also work out what arePin(2, 0), Pin(1, 1), and Pin(0, 2); see Choquet-Bruhat [11], Chapter I, Section 7, page 26.Show that

Spin(0, 2) = Spin(2, 0) ≈ U(1),

andSpin(1, 1) = {a1 + be1e2 | a2 − b2 = 1}.

Observe that Spin(1, 1) is not connected.

More generally, it can be shown that Cl0p,q and Cl0q,p are isomorphic, from which it followsthat Spin(p, q) and Spin(q, p) are isomorphic, but Pin(p, q) andPin(q, p) are not isomorphicin general, and in particular, Pin(p, 0) and Pin(0, p) are not isomorphic in general (seeChoquet-Bruhat [11], Chapter I, Section 7). However, due to the “8-periodicity” of theClifford algebras (to be discussed in the next section), it follows that Clp,q and Clq,p areisomorphic when |p− q| = 0 mod 4.

1.6 Periodicity of the Clifford Algebras Clp,q

It turns out that the real algebras Clp,q can be build up as tensor products of the basicalgebras R, C, and H. As pointed out by Lounesto (Section 23.16 [21]), the description ofthe real algebras Clp,q as matrix algebras and the 8-periodicity was first observed by ElieCartan in 1908; see Cartan’s article, Nombres Complexes, based on the original article inGerman by E. Study, in Molk [23], article I-5 (fasc. 3), pages 329-468. These algebras aredefined in Section 36 under the name “‘Systems of Clifford and Lipschitz,” page 463-466.Of course, Cartan used a very different notation; see page 464 in the article cited above.These facts were rediscovered independently by Raoul Bott in the 1960’s (see Raoul Bott’scomments in Volume 2 of his Collected papers.).

We will use the notation R(n) (resp. C(n)) for the algebra Mn(R) of all n × n realmatrices (resp. the algebra Mn(C) of all n×n complex matrices). As mentioned in Example1.3, it is not hard to show that

Cl0,1 = C Cl1,0 = R⊕ R

Cl0,2 = H Cl2,0 = R(2),


andCl1,1 = R(2).

The key to the classification is the following lemma:

Lemma 1.15. We have the isomorphisms

Cl0,n+2 ≈ Cln,0 ⊗ Cl0,2

Cln+2,0 ≈ Cl0,n ⊗ Cl2,0

Clp+1,q+1 ≈ Clp,q ⊗ Cl1,1,

for all n, p, q ≥ 0.

Proof. Let Φ0,n(x) = −‖x‖2, where ‖x‖ is the standard Euclidean norm on Rn+2, and let(e1, . . . , en+2) be an orthonormal basis for Rn+2 under the standard Euclidean inner product.We also let (e′1, . . . , e

′n) be a set of generators for Cln,0 and (e′′1, e

′′2) be a set of generators

for Cl0,2. We can define a linear map f : Rn+2 → Cln,0 ⊗ Cl0,2 by its action on the basis(e1, . . . , en+2) as follows:

f(ei) =

{e′i ⊗ e′′1e

′′2 for 1 ≤ i ≤ n

1⊗ e′′i−n for n+ 1 ≤ i ≤ n+ 2.

Observe that for 1 ≤ i, j ≤ n, we have

f(ei)f(ej) + f(ej)f(ei) = (e′ie′j + e′je

′i)⊗ (e′′1e

′′e)

2 = −2δij1⊗ 1,

since e′′1e′′2 = −e′′2e

′′1, (e

′′1)

2 = −1, and (e′′2)2 = −1, and e′ie

′j = −e′je

′i, for all i 6= j, and

(e′i)2 = 1, for all i with 1 ≤ i ≤ n. Also, for n + 1 ≤ i, j ≤ n+ 2, we have

f(ei)f(ej) + f(ej)f(ei) = 1⊗ (e′′i−ne′′j−n + e′′j−ne

′′i−n) = −2δij1⊗ 1,

andf(ei)f(ek) + f(ek)f(ei) = 2e′i ⊗ (e′′1e

′′2e

′′n−k + e′′n−ke

′′1e

′′2) = 0,

for 1 ≤ i, j ≤ n and n+ 1 ≤ k ≤ n+ 2 (since e′′n−k = e′′1 or e′′n−k = e′′2). Thus, we have

f(x)2 = −‖x‖2 · 1⊗ 1 for all x ∈ Rn+2,

and by the universal mapping property of Cl0,n+2, we get an algebra map

f : Cl0,n+2 → Cln,0 ⊗ Cl0,2.

Since f maps onto a set of generators, it is surjective. However

dim(Cl0,n+2) = 2n+2 = 2n · 2 = dim(Cln,0)dim(Cl0,2) = dim(Cln,0 ⊗ Cl0,2),

and f is an isomorphism.

1.6. PERIODICITY OF THE CLIFFORD ALGEBRAS CLP,Q 35

The proof of the second identity is analogous. For the third identity, we have

Φp,q(x1, . . . , xp+q) = x21 + · · ·+ x2

p − (x2p+1 + · · ·+ x2

p+q),

and let (e1, . . . , ep+1, ǫ1, . . . , ǫq+1) be an orthogonal basis for Rp+q+2 so that Φp+1,q+1(ei) = +1and Φp+1,q+1(ǫj) = −1 for i = 1, . . . , p+1 and j = 1, . . . , q+1. Also, let (e′1, . . . , e

′p, ǫ

′1, . . . , ǫ

′q)

be a set of generators for Clp,q and (e′′1, ǫ′′1) be a set of generators for Cl1,1. We define a linear

map f : Rp+q+2 → Clp,q ⊗ Cl1,1 by its action on the basis as follows:

f(ei) =

{e′i ⊗ e′′1ǫ

′′1 for 1 ≤ i ≤ p

1⊗ e′′1 for i = p+ 1,

and

f(ǫj) =

{ǫ′j ⊗ e′′1ǫ

′′1 for 1 ≤ j ≤ q

1⊗ ǫ′′1 for j = q + 1.

We can check that

f(x)2 = Φp+1,q+1(x) · 1⊗ 1 for all x ∈ Rp+q+2,

and we finish the proof as in the first case.

To apply this lemma, we need some further isomorphisms among various matrix algebras.

Proposition 1.16. The following isomorphisms hold:

R(m)⊗ R(n) ≈ R(mn) for all m,n ≥ 0

R(n)⊗R K ≈ K(n) for K = C or K = H and all n ≥ 0

C⊗R C ≈ C⊕ C

C⊗R H ≈ C(2)

H⊗R H ≈ R(4).

Proof. Details can be found in Lawson and Michelsohn [20]. The first two isomorphisms arequite obvious. The third isomorphism C⊕ C → C⊗ C is obtained by sending

(1, 0) 7→ 1

2(1⊗ 1 + i⊗ i), (0, 1) 7→ 1

2(1⊗ 1− i⊗ i).

The field C is isomorphic to the subring of H generated by i. Thus, we can view H as aC-vector space under left scalar multiplication. Consider the R-bilinear mapπ : C×H → HomC(H,H) given by

πy,z(x) = yxz,

where y ∈ C and x, z ∈ H. Thus, we get an R-linear map π : C ⊗R H → HomC(H,H).However, we have HomC(H,H) ≈ C(2). Furthermore, since

πy,z ◦ πy′,z′ = πyy′,zz′,


the map π is an algebra homomorphism

π : C×H → C(2).

We can check on a basis that π is injective, and since

dimR(C×H) = dimR(C(2)) = 8,

the map π is an isomorphism. The last isomorphism is proved in a similar fashion.

We now have the main periodicity theorem.

Theorem 1.17. (Cartan/Bott) For all n ≥ 0, we have the following isomorphisms:

Cl0,n+8 ≈ Cl0,n ⊗ Cl0,8

Cln+8,0 ≈ Cln,0 ⊗ Cl8,0.

Furthermore,Cl0,8 = Cl8,0 = R(16).

Proof. By Lemma 1.15 we have the isomorphisms

Cl0,n+2 ≈ Cln,0 ⊗ Cl0,2

Cln+2,0 ≈ Cl0,n ⊗ Cl2,0,

and thus,

Cl0,n+8 ≈ Cln+6,0 ⊗Cl0,2 ≈ Cl0,n+4 ⊗Cl2,0 ⊗Cl0,2 ≈ · · · ≈ Cl0,n ⊗Cl2,0 ⊗Cl0,2 ⊗Cl2,0 ⊗Cl0,2.

Since Cl0,2 = H and Cl2,0 = R(2), by Proposition 1.16, we get

Cl2,0 ⊗ Cl0,2 ⊗ Cl2,0 ⊗ Cl0,2 ≈ H⊗H⊗ R(2)⊗ R(2) ≈ R(4)⊗ R(4) ≈ R(16).

The second isomorphism is proved in a similar fashion.

From all this, we can deduce the following table:

n 0 1 2 3 4 5 6 7 8Cl0,n R C H H⊕H H(2) C(4) R(8) R(8)⊕ R(8) R(16)Cln,0 R R⊕ R R(2) C(2) H(2) H(2)⊕H(2) H(4) C(8) R(16)

A table of the Clifford groups Clp,q for 0 ≤ p, q ≤ 7 can be found in Kirillov [18], and for0 ≤ p, q ≤ 8, in Lawson and Michelsohn [20] (but beware that their Clp,q is our Clq,p). It canalso be shown that

Clp+1,q ≈ Clq+1,p

Clp,q ≈ Clp−4,q+4

1.7. THE COMPLEX CLIFFORD ALGEBRAS CL(N,C) 37

with p ≥ 4 in the second identity (see Lounesto [21], Chapter 16, Sections 16.3 and 16.4).Using the second identity, if |p−q| = 4k, it is easily shown by induction on k that Clp,q ≈ Clq,p,as claimed in the previous section.

We also have the isomorphisms

Clp,q ≈ Cl0p,q+1,

frow which it follows thatSpin(p, q) ≈ Spin(q, p)

(see Choquet-Bruhat [11], Chapter I, Sections 4 and 7). However, in general, Pin(p, q) andPin(q, p) are not isomorphic. In fact, Pin(0, n) and Pin(n, 0) are not isomorphic if n 6= 4k,with k ∈ N (see Choquet-Bruhat [11], Chapter I, Section 7, page 27).

1.7 The Complex Clifford Algebras Cl(n,C)

One can also consider Clifford algebras over the complex field C. In this case, it is well-knownthat every nondegenerate quadratic form can be expressed by

ΦC

n(x1, . . . , xn) = x21 + · · ·+ x2

n

in some orthonormal basis. Also, it is easily shown that the complexification C ⊗R Clp,q ofthe real Clifford algebra Clp,q is isomorphic to Cl(ΦC

n). Thus, all these complex algebras areisomorphic for p+q = n, and we denote them by Cl(n,C). Theorem 1.15 yields the followingperiodicity theorem:

Theorem 1.18. The following isomorphisms hold:

Cl(n+ 2,C) ≈ Cl(n,C)⊗C Cl(2,C),

with Cl(2,C) = C(2).

Proof. Since Cl(n,C) = C⊗R Cl0,n = C⊗R Cln,0, by Lemma 1.15, we have

Cl(n+ 2,C) = C⊗R Cl0,n+2 ≈ C⊗R (Cln,0 ⊗R Cl0,2) ≈ (C⊗R Cln,0)⊗C (C⊗R Cl0,2).

However, Cl0,2 = H, Cl(n,C) = C ⊗R Cln,0, and C⊗R H ≈ C(2), so we get Cl(2,C) = C(2)and

Cl(n+ 2,C) ≈ Cl(n,C)⊗C C(2),

and the theorem is proved.

As a corollary of Theorem 1.18, we obtain the fact that

Cl(2k,C) ≈ C(2k) and Cl(2k + 1,C) ≈ C(2k)⊕ C(2k).


The table of the previous section can also be completed as follows:

n 0 1 2 3 4 5 6 7 8Cl0,n R C H H⊕H H(2) C(4) R(8) R(8)⊕ R(8) R(16)Cln,0 R R⊕ R R(2) C(2) H(2) H(2)⊕H(2) H(4) C(8) R(16)

Cl(n,C) C 2C C(2) 2C(2) C(4) 2C(4) C(8) 2C(8) C(16).

where 2C(k) is an abbrevation for C(k)⊕ C(k).

1.8 The Groups Pin(p, q) and Spin(p, q) as double covers

of O(p, q) and SO(p, q)

It turns out that the groups Pin(p, q) and Spin(p, q) have nice topological properties w.r.t.the groups O(p, q) and SO(p, q). To explain this, we review the definition of covering mapsand covering spaces (for details, see Fulton [14], Chapter 11). Another interesting source isChevalley [9], where is is proved that Spin(n) is a universal double cover of SO(n) for alln ≥ 3.

Since Cp,q is an algebra of dimension 2p+q, it is a topological space as a vector spaceisomorphic to V = R2p+q

. Now, the group C∗p,q of units of Cp,q is open in Cp,q, because

x ∈ Cp,q is a unit if the linear map y 7→ xy is an isomorphism, and GL(V ) is open inEnd(V ), the space of endomorphisms of V . Thus, C∗

p,q is a Lie group, and since Pin(p, q)and Spin(p, q) are clearly closed subgroups of C∗

p,q, they are also Lie groups.

Definition 1.7. Given two topological spaces X and Y , a covering map is a continuoussurjective map p : Y → X with the property that for every x ∈ X , there is some open subsetU ⊆ X with x ∈ U , so that p−1(U) is the disjoint union of open subsets Vα ⊆ Y , and therestriction of p to each Vα is a homeomorphism onto U . We say that U is evenly coveredby p. We also say that Y is a covering space of X . A covering map p : Y → X is calledtrivial if X itself is evenly covered by p (i.e., Y is the disjoint union of open subsets Yα eachhomeomorphic to X), and nontrivial otherwise. When each fiber p−1(x) has the same finitecardinaly n for all x ∈ X , we say that p is an n-covering (or n-sheeted covering).

Note that a covering map p : Y → X is not always trivial, but always locally trivial (i.e.,for every x ∈ X , it is trivial in some open neighborhood of x). A covering is trivial iff Yis isomorphic to a product space of X × T , where T is any set with the discrete topology.Also, if Y is connected, then the covering map is nontrivial.

Definition 1.8. An isomorphism ϕ between covering maps p : Y → X and p′ : Y ′ → X is ahomeomorphism ϕ : Y → Y ′ so that p = p′ ◦ ϕ.

Typically, the space X is connected, in which case it can be shown that all the fibersp−1(x) have the same cardinality.

One of the most important properties of covering spaces is the path–lifting property, aproperty that we will use to show that Spin(n) is path-connected.

1.8. THE GROUPS PIN(P,Q) AND SPIN(P,Q) AS DOUBLE COVERS 39

Proposition 1.19. (Path lifting) Let p : Y → X be a covering map, and let γ : [a, b] → Xbe any continuous curve from xa = γ(a) to xb = γ(b) in X. If y ∈ Y is any point so thatp(y) = xa, then there is a unique curve γ : [a, b] → Y so that y = γ(a) and

p ◦ γ(t) = γ(t) for all t ∈ [a, b].

Proof. See Fulton [15], Chapter 11, Lemma 11.6.

Many important covering maps arise from the action of a group G on a space Y . If Yis a topological space, an action (on the left) of a group G on Y is a map α : G × Y → Ysatisfying the following conditions, where for simplicity of notation, we denote α(g, y) byg · y:

(1) g · (h · y) = (gh) · y, for all g, h ∈ G and y ∈ Y ;

(2) 1 · y = y, for all ∈ Y , where 1 is the identity of the group G;

(3) The map y 7→ g · y is a homeomorphism of Y for every g ∈ G.

We define an equivalence relation on Y as follows: x ≡ y iff y = g · x for some g ∈ G(check that this is an equivalence relation). The equivalence class G · x = {g · x | g ∈ G} ofany x ∈ Y is called the orbit of x. We obtain the quotient space Y/G and the projectionmap p : Y → Y/G sending every y ∈ Y to its orbit. The space Y/G is given the quotienttopology (a subset U of Y/G is open iff p−1(U) is open in Y ).

Given a subset V of Y and any g ∈ G, we let

g · V = {g · y | y ∈ V }.

We say that G acts evenly on Y if for every y ∈ Y , there is an open subset V containing yso that g · V and h · V are disjoint for any two distinct elements g, h ∈ G.

The importance of the notion a group acting evenly is that such actions induce a coveringmap.

Proposition 1.20. If G is a group acting evenly on a space Y , then the projection mapp : Y → Y/G is a covering map.

Proof. See Fulton [15], Chapter 11, Lemma 11.17.

The following proposition shows that Pin(p, q) and Spin(p, q) are nontrivial coveringspaces, unless p = q = 1.

Proposition 1.21. For all p, q ≥ 0, the groups Pin(p, q) and Spin(p, q) are double covers ofO(p, q) and SO(p, q), respectively. Furthermore, they are nontrivial covers unless p = q = 1.


Proof. We know that kernel of the homomorphism ρ : Pin(p, q) → O(p, q) is Z2 = {−1, 1}.If we let Z2 act on Pin(p, q) in the natural way, then O(p, q) ≈ Pin(p, q)/Z2, and the readercan easily check that Z2 acts evenly. By Proposition 1.20, we get a double cover. Theargument for ρ : Spin(p, q) → SO(p, q) is similar.

Let us now assume that p 6= 1 or q 6= 1. In order to prove that we have nontrivialcovers, it is enough to show that −1 and 1 are connected by a path in Pin(p, q) (If we hadPin(p, q) = U1 ∪ U2 with U1 and U2 open, disjoint, and homeomorphic to O(p, q), then −1and 1 would not be in the same Ui, and so, they would be in disjoint connected components.Thus, −1 and 1 can’t be path–connected, and similarly with Spin(p, q) and SO(p, q).) Since(p, q) 6= (1, 1), we can find two orthogonal vectors e1 and e2 so that Φp,q(e1) = Φp,q(e2) = ±1.Then,

γ(t) = ± cos(2t) 1 + sin(2t) e1e2 = (cos t e1 + sin t e2)(sin t e2 − cos t e1),

for 0 ≤ t ≤ π, defines a path in Spin(p, q), since

(± cos(2t) 1 + sin(2t) e1e2)−1 = ± cos(2t) 1− sin(2t) e1e2,

as desired.

In particular, if n ≥ 2, since the group SO(n) is path-connected, the group Spin(n) isalso path-connected. Indeed, given any two points xa and xb in Spin(n), there is a pathγ from ρ(xa) to ρ(xb) in SO(n) (where ρ : Spin(n) → SO(n) is the covering map). ByProposition 1.19, there is a path γ in Spin(n) with origin xa and some origin xb so thatρ(xb) = ρ(xb). However, ρ−1(ρ(xb)) = {−xb, xb}, and so xb = ±xb. The argument used inthe proof of Proposition 1.21 shows that xb and −xb are path-connected, and so, there is apath from xa to xb, and Spin(n) is path-connected.

In fact, for n ≥ 3, it turns out that Spin(n) is simply connected. Such a covering spaceis called a universal cover (for instance, see Chevalley [9]).

This last fact requires more algebraic topology than we are willing to explain in detail,and we only sketch the proof. The notions of fibre bundle, fibration, and homotopy sequenceassociated with a fibration are needed in the proof. We refer the perseverant readers to Bottand Tu [5] (Chapter 1 and Chapter 3, Sections 16–17) or Rotman [25] (Chapter 11) for adetailed explanation of these concepts.

Recall that a topological space is simply connected if it is path connected and if π1(X) =(0), which means that every closed path in X is homotopic to a point. Since we just provedthat Spin(n) is path connected for n ≥ 2, we just need to prove that π1(Spin(n)) = (0) forall n ≥ 3. The following facts are needed to prove the above assertion:

(1) The sphere Sn−1 is simply connected for all n ≥ 3.

(2) The group Spin(3) ≃ SU(2) is homeomorphic to S3, and thus, Spin(3) is simplyconnected.

1.8. THE GROUPS PIN(P,Q) AND SPIN(P,Q) AS DOUBLE COVERS 41

(3) The group Spin(n) acts on Sn−1 in such a way that we have a fibre bundle with fibreSpin(n− 1):

Spin(n− 1) −→ Spin(n) −→ Sn−1.

Fact (1) is a standard proposition of algebraic topology, and a proof can found in manybooks. A particularly elegant and yet simple argument consists in showing that any closedcurve on Sn−1 is homotopic to a curve that omits some point. First, it is easy to see thatin Rn, any closed curve is homotopic to a piecewise linear curve (a polygonal curve), andthe radial projection of such a curve on Sn−1 provides the desired curve. Then, we use thestereographic projection of Sn−1 from any point omitted by that curve to get another closedcurve in Rn−1. Since Rn−1 is simply connected, that curve is homotopic to a point, and so isits preimage curve on Sn−1. Another simple proof uses a special version of the Seifert—vanKampen’s theorem (see Gramain [17]).

Fact (2) is easy to establish directly, using (1).

To prove (3), we let Spin(n) act on Sn−1 via the standard action: x ·v = xvx−1. BecauseSO(n) acts transitively on Sn−1 and there is a surjection Spin(n) −→ SO(n), the groupSpin(n) also acts transitively on Sn−1. Now, we have to show that the stabilizer of anyelement of Sn−1 is Spin(n− 1). For example, we can do this for e1. This amounts to somesimple calculations taking into account the identities among basis elements. Details of thisproof can be found in Mneimne and Testard [22], Chapter 4. It is still necessary to prove thatSpin(n) is a fibre bundle over Sn−1 with fibre Spin(n − 1). For this, we use the followingresults whose proof can be found in Mneimne and Testard [22], Chapter 4:

Lemma 1.22. Given any topological group G, if H is a closed subgroup of G and the pro-jection π : G → G/H has a local section at every point of G/H, then

H −→ G −→ G/H

is a fibre bundle with fibre H.

Lemma 1.22 implies the following key proposition:

Proposition 1.23. Given any linear Lie group G, if H is a closed subgroup of G, then

H −→ G −→ G/H

is a fibre bundle with fibre H.

Now, a fibre bundle is a fibration (as defined in Bott and Tu [5], Chapter 3, Section 16,or in Rotman [25], Chapter 11). For a proof of this fact, see Rotman [25], Chapter 11, orMneimne and Testard [22], Chapter 4. So, there is a homotopy sequence associated withthe fibration (Bott and Tu [5], Chapter 3, Section 17, or Rotman [25], Chapter 11, Theorem11.48), and in particular, we have the exact sequence

π1(Spin(n− 1)) −→ π1(Spin(n)) −→ π1(Sn−1).


Since π1(Sn−1) = (0) for n ≥ 3, we get a surjection

π1(Spin(n− 1)) −→ π1(Spin(n)),

and so, by induction and (2), we get

π1(Spin(n)) ≈ π1(Spin(3)) = (0),

proving that Spin(n) is simply connected for n ≥ 3.

We can also show that π1(SO(n)) = Z/2Z for all n ≥ 3. For this, we use Theorem 1.11and Proposition 1.21, which imply that Spin(n) is a fibre bundle over SO(n) with fibre{−1, 1}, for n ≥ 2:

{−1, 1} −→ Spin(n) −→ SO(n).

Again, the homotopy sequence of the fibration exists, and in particular we get the exactsequence

π1(Spin(n)) −→ π1(SO(n)) −→ π0({−1,+1}) −→ π0(SO(n)).

Since π0({−1,+1}) = Z/2Z, π0(SO(n)) = (0), and π1(Spin(n)) = (0), when n ≥ 3, we getthe exact sequence

(0) −→ π1(SO(n)) −→ Z/2Z −→ (0),

and so, π1(SO(n)) = Z/2Z. Therefore, SO(n) is not simply connected for n ≥ 3.

Remark: Of course, we have been rather cavalier in our presentation. Given a topologicalspace X , the group π1(X) is the fundamental group of X , i.e. the group of homotopy classesof closed paths in X (under composition of loops). But π0(X) is generally not a group!Instead, π0(X) is the set of path-connected components of X . However, when X is a Liegroup, π0(X) is indeed a group. Also, we have to make sense of what it means for thesequence to be exact. All this can be made rigorous (see Bott and Tu [5], Chapter 3, Section17, or Rotman [25], Chapter 11).

1.9 More on the Topology of O(p, q) and SO(p, q)

It turns out that the topology of the group O(p, q) is completely determined by the topologyof O(p) and O(q). This result can be obtained as a simple consequence of some standardLie group theory. The key notion is that of a pseudo-algebraic group.

Consider the group GL(n,C) of invertible n × n matrices with complex coefficients. IfA = (akl) is such a matrix, denote by xkl the real part (resp. ykl, the imaginary part) of akl(so, akl = xkl + iykl).

Definition 1.9. A subgroup G of GL(n,C) is pseudo-algebraic iff there is a finite set ofpolynomials in 2n2 variables with real coefficients {Pi(X1, . . . , Xn2, Y1, . . . , Yn2)}ti=1, so that

A = (xkl + iykl) ∈ G iff Pi(x11, . . . , xnn, y11, . . . , ynn) = 0, for i = 1, . . . , t.

1.9. MORE ON THE TOPOLOGY OF O(P,Q) AND SO(P,Q) 43

Recall that if A is a complex n × n-matrix, its adjoint A∗ is defined by A∗ = (A)⊤.Also, U(n) denotes the group of unitary matrices, i.e., those matrices A ∈ GL(n,C) sothat AA∗ = A∗A = I, and H(n) denotes the vector space of Hermitian matrices, i.e., thosematrices A so that A∗ = A. Then, we have the following theorem which is essentially arefined version of the polar decomposition of matrices:

Theorem 1.24. Let G be a pseudo-algebraic subgroup of GL(n,C) stable under adjunction(i.e., we have A∗ ∈ G whenever A ∈ G). Then, there is some integer d ∈ N so that G ishomeomorphic to (G ∩U(n))× Rd. Moreover, if g is the Lie algebra of G, the map

(U(n) ∩G)× (H(n) ∩ g) −→ G, given by (U,H) 7→ UeH ,

is a homeomorphism onto G.

Proof. A proof can be found in Knapp [19], Chapter 1, or Mneimne and Testard [22], Chapter3.

We now apply Theorem 1.24 to determine the structure of the space O(p, q). Let Jp,q bethe matrix

Jp,q =

(Ip 00 −Iq

).

We know that O(p, q) consists of the matrices A in GL(p+ q,R) such that

A⊤Jp,qA = Jp,q,

and so O(p, q) is clearly pseudo-algebraic. Using the above equation, it is easy to determinethe Lie algebra, o(p, q), of O(p, q). We find that o(p, q) is given by

o(p, q) =

{(X1 X2

X⊤2 X3

) ∣∣∣∣ X⊤1 = −X1, X⊤

3 = −X3, X2 arbitrary

},

where X1 is a p× p matrix, X3 is a q× q matrix, and X2 is a p× q matrix. Consequently, itimmediately follows that

o(p, q) ∩H(p+ q) =

{(0 X2

X⊤2 0

) ∣∣∣∣ X2 arbitrary

},

a vector space of dimension pq.

Some simple calculations also show that

O(p, q) ∩U(p+ q) =

{(X1 00 X2

) ∣∣∣∣ X1 ∈ O(p), X2 ∈ O(q)

}∼= O(p)×O(q).

Therefore, we obtain the structure of O(p, q):


Proposition 1.25. The topological space O(p, q) is homeomorphic to O(p)×O(q)× Rpq.

Since O(p) has two connected components when p ≥ 1, we see that O(p, q) has fourconnected components when p, q ≥ 1. It is also obvious that

SO(p, q) ∩U(p+ q) =

{(X1 00 X2

) ∣∣∣∣ X1 ∈ O(p), X2 ∈ O(q), det(X1) det(X2) = 1

}.

This is a subgroup of O(p)×O(q) that we denote S(O(p)×O(q)). Furthermore, it is easyto show that so(p, q) = o(p, q). Thus, we also have

Proposition 1.26. The topological space SO(p, q) is homeomorphic to S(O(p)×O(q))×Rpq.

Note that SO(p, q) has two connected components when p, q ≥ 1. The connectedcomponent of Ip+q is a group denoted SO0(p, q). This latter space is homeomorphic toSO(p)× SO(q)× Rpq.

As a closing remark observe that the dimension of all these spaces depends only on p+ q:It is (p+ q)(p+ q − 1)/2.

Acknowledgments. I thank Eric King whose incisive questions and relentless quest forthe “essence” of rotations eventually caused a level of discomfort high enough to force meto improve the clarity of these notes. Rotations are elusive! I also thank: Fred Lunnonfor many insighful comments and for catching a number of typos; and Troy Woo who alsoreported some typos.

Bibliography

[1] Michael Artin. Algebra. Prentice Hall, first edition, 1991.

[2] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. AddisonWesley, third edition, 1969.

[3] Michael F Atiyah, Raoul Bott, and Arnold Shapiro. Clifford modules. Topology, 3,Suppl. 1:3–38, 1964.

[4] Andrew Baker. Matrix Groups. An Introduction to Lie Group Theory. SUMS. Springer,2002.

[5] Raoul Bott and Tu Loring W. Differential Forms in Algebraic Topology. GTM No. 82.Springer Verlag, first edition, 1986.

[6] Nicolas Bourbaki. Algebre, Chapitre 9. Elements de Mathematiques. Hermann, 1968.

[7] T. Brocker and T. tom Dieck. Representation of Compact Lie Groups. GTM, Vol. 98.Springer Verlag, first edition, 1985.

[8] Elie Cartan. Theory of Spinors. Dover, first edition, 1966.

[9] Claude Chevalley. Theory of Lie Groups I. Princeton Mathematical Series, No. 8.Princeton University Press, first edition, 1946. Eighth printing.

[10] Claude Chevalley. The Algebraic Theory of Spinors and Clifford Algebras. CollectedWorks, Vol. 2. Springer, first edition, 1997.

[11] Yvonne Choquet-Bruhat and Cecile DeWitt-Morette. Analysis, Manifolds, and Physics,Part II: 92 Applications. North-Holland, first edition, 1989.

[12] Morton L. Curtis. Matrix Groups. Universitext. Springer Verlag, second edition, 1984.

[13] Jean Dieudonne. Sur les Groupes Classiques. Hermann, third edition, 1967.

[14] William Fulton. Algebraic Topology, A first course. GTM No. 153. Springer Verlag, firstedition, 1995.

45

46 BIBLIOGRAPHY

[15] William Fulton and Joe Harris. Representation Theory, A first course. GTM No. 129.Springer Verlag, first edition, 1991.

[16] Jean H. Gallier. Geometric Methods and Applications, For Computer Science and En-gineering. TAM, Vol. 38. Springer, second edition, 2011.

[17] Andre Gramain. Topologie des Surfaces. Collection Sup. Puf, first edition, 1971.

[18] A.A. Kirillov. Spinor representations of orthogonal groups. Technical report, Universityof Pennsylvania, Math. Department, Philadelphia, PA 19104, 2001. Course Notes forMath 654.

[19] Anthony W. Knapp. Lie Groups Beyond an Introduction. Progress in Mathematics,Vol. 140. Birkhauser, second edition, 2002.

[20] Blaine H. Lawson and Marie-Louise Michelsohn. Spin Geometry. Princeton Math.Series, No. 38. Princeton University Press, 1989.

[21] Pertti Lounesto. Clifford Algebras and Spinors. LMS No. 286. Cambridge UniversityPress, second edition, 2001.

[22] R. Mneimne and F. Testard. Introduction a la Theorie des Groupes de Lie Classiques.Hermann, first edition, 1997.

[23] Jules Molk. Encyclopedie des Sciences Mathematiques Pures et Appliquees. Tome I(premier volume), Arithmetique. Gauthier-Villars, first edition, 1916.

[24] Ian R. Porteous. Topological Geometry. Cambridge University Press, second edition,1981.

[25] Joseph J. Rotman. Introduction to Algebraic Topology. GTM No. 119. Springer Verlag,first edition, 1988.

Documents

Clifford Algebras, Clifford Groups, and a Generalization of the ... · Chapter 1 Clifford Algebras, Clifford Groups, and the Groups Pin(n) and Spin(n) 1.1 Introduction: Rotations